Invert about the point A, with arbitrary radius. You should end up with a picture that looks the same as the original one, but flipped, i.e. similar but oppositely oriented. Then, from the co-linearity of A,Q,(image of Q) and A,P,(image of P) and the fact that (image of P) corresponds to Q and (image of Q) corresponds to P under the similarity, angle PAO = angle QAO